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Full text of "Treatise On Analysis Vol-Ii"

150       XIII    INTEGRATION

15.    Let / be a lower (or upper) semicontinuous real-valued function on Rn. Let X be a
locally compact space and ui9..., un universally measurable finite real-valued functions.
Show that/(#i,..., un) is universally measurable (cf. Problem 17).

16.    With the notation of Section 12.7, Problem 3, show that there exist n universally
measurable mappings t\—>zk(t) of C" into C (1 ^ k <[ n) such that

Pt(X) = X" + hX"-1 + ••• + *„ - fl (X ~ **(0).
(By induction on the degree Ğ, using Problem 15.)

17.    Let / be the continuous function on I = [0, 1 ] which was defined in Section 4.2,
Problem 2(d), such that the restriction of/to the Cantor set K is a bijection of K onto
I. Let g be the function defined by g(x) = x +/(jc); then g is a homeomorphism of I
onto 21 = [0,2], such that g(K) is a compact set of measure 1  (with respect to
Lebesgue measure), although K is negligible. If h(x) = g~1(2x), then h is a homeo-
morphism of I onto I such that h~^(K^ is not negligible with respect to Lebesgue
measure. Moreover, if A is a nonmeasurable set (with respect to Lebesgue measure)
contained in #(K), then the set B = /z(J-A) is negligible and therefore measurable. The
function <pB ° h is not measurable, although h is continuous and <pB is measurable
(cf. (16.23)).

18.    (a)   Let/be a finite continuous real-valued function on R. Show that there exists a
real-valued function g on R which is universally measurable and such that/(#(*)) = x
for all x e/(R) (cf. Section 12.7, Problem 1).

(b) Assume that there exists a partition of I == [0,1] into a family (Hn)neZ of non-
measurable sets (with respect to Lebesgue measure) such that Card (Hn) = Card(R)
for all neZ. Let K be the Cantor set (Section 4.2, Problem 2) and F = (J (K + Ğ),

n e Z

so that F is a negligible closed set. Show that there exists a bijection/: R-^R such
that/(F n ]n, n + 1 [) = Hn for all n, and such that / is linear in each of the com-
ponent intervals of OF- Then / is measurable with respect to Lebesgue measure, but
f~l is not.

19.    Let X be a locally compact space and ft a positive measure on X. Let D be a de-
numerable dense subset of R. Show that a mapping/: X-ğR is ft-measurable if and
only if, for each r E D, the set of points x e X such that/(x) ^ r is ft-measurable.

20.   Suppose ft bounded and ft(X) = 1. If /^ 0 is an integrable function, show that
(1 +/2)1/2 is integrable and that, if A = (fdp,, then

(1 + A2)1/2 g J(l +/2)1/2 dfji g 1 + A.

(Use Problem 14(a) of Section 13.8.)

21. Let X, Y, Z be three locally compact spaces, if /: X -> Y and g : Y -+ Z are universally
measurable, show that g of is universally measurable. (To show that# °/ is/^-measur-
able where ^ is any positive measure on X, reduce to the case where X is compact and
consider the measure/(/x) on Y (Section 13.4, Problem 8).) In particular, if B <= Y is
universally measurable, then/-1(B) is universally measurable.